Controllability of complex networks: Choosing the best driver set

Identifying the best driver set in a complex network is an unsolved problem in the application of pinning control methods. We choose the eigenratio of the augmented Laplacian matrix, and the best driver set is the subset of nodes providing the most effective pinning control strategy, i.e. the one for which synchronization of the whole network to the reference state is attained over the widest range of the coupling parameter. In this work, we propose a centrality measure based on sensitivity analysis of the Laplacian matrix of the connection graph to find an approximate solution to this problem. The proposed metric is computationally efficient as it requires only a single eigen-decomposition of the Laplacian matrix. Numerical results on a number of sample networks show that the proposed metric has significantly better accuracy than currently used heuristics, and in most cases can correctly identify the true optimal set, which is obtainable through combinatorial search.